Projective Planes of Order 49 Related to t66

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t66 and all known planes of order 49 obtained from it by dualizing and deriving. Coming soon: also planes related by the method of lifting quotients. This list is currently incomplete; check back later for a complete enumeration.

Following the table is a key to the table.

Known Projective Planes of Order 49 Related to t66

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t66, dual dt66	172872	2⁴,3⁴,6³,12,2401	1,98⁴,147⁴,294³,588	939
2	t66_0_0, dt66_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t66_0_1, dt66_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t66_0_2, dt66_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t66_0_3, dt66_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t66_0_4, dt66_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t66_0_5, dt66_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t66_0_6, dt66_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t66_0_7, dt66_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t66_1_0, dt66_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t66_1_1, dt66_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t66_1_2, dt66_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t66_1_3, dt66_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t66_2_0, dt66_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t66_2_1, dt66_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t66_2_2, dt66_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t66_2_3, dt66_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t66_3_0, dt66_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t66_3_1, dt66_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t66_3_2, dt66_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t66_3_3, dt66_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t66_4_0, dt66_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t66_4_1, dt66_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t66_5_0, dt66_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t66_5_1, dt66_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t66_6_0, dt66_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t66_6_1, dt66_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t66_7_0, dt66_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t66_7_1, dt66_7_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t66_8_0, dt66_8_0	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
31	t66_8_1, dt66_8_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t66_9_0, dt66_9_0	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
33	t66_9_1, dt66_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t66_10_0, dt66_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t66_10_1, dt66_10_1	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
36	t66_11_0, dt66_11_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t66_11_1, dt66_11_1	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987

Key to the table

Only one line is displayed for both a plane and its dual, an asterisk (*) in the first column indicating that the plane is self-dual. Each line includes the following information and isomorphism invariants for each plane.

Plane provides a pzip file containing the projective plane. I assume you have installed pzip under linux, which compresses each plane to about 6 KB. Right-click to save as plane.pz, then type the command punzip plane.pz to recover plane as a text file. This text file has 2451 rows, each row specifying a line of the plane as a subset of the points 0,1,2,...,2450. For non-self-dual planes, a second file is also given, containing the dual of the first.
|Autgp| The order of the full collineation group of the projective plane. This table entry is linked to a gzip file providing generators of the automorphism group. Right-click to save as plane.gz, then type the command gunzip plane.gz to recover plane as a text file. This text file lists generators of the full collineation group as permutations of the integers 0,1,2,...,4901 where 0,1,2,...,2450 are labels for the points and 2451,...,4901 are labels for the lines. Generators for the automorphism group of the dual plane are not provided since these are trivially obtained from the generators given for the original plane.
Point Orbits The lengths of the full automorphism group on the points. (These are the lengths of the line orbits for the dual plane.)
Line Orbits The lengths of the full automorphism group on the lines. (These are the lengths of the point orbits for the dual plane.)
7-rank The rank of the (0,1)-incidence matrix of the projective plane, over a field of characteristic 7.

/ revised February, 2011